Hodgson’s England deal expires after the European Championships and FA chief executive Martin Glenn has confirmed that his future will be decided on how the team perform in France next summer.

Anything less than a quarter-final appearance is likely to see the FA make a change with the governing body ready to widen the search beyond just English candidates, as was the case when Hodgson was appointed in 2012.

FA chairman Greg Dyke has said a foreign manager could succeed Hodgson, but it is understood that the next man will at least preferably fulfil a ‘home-grown’ criteria.

That opens the door to Northern Irishman Rodgers, Spaniard Roberto Martinez and even Scotsman David Moyes. Arsene Wenger would no doubt be the FA’s dream appointment, but the Frenchman has turned down the England job in the past.

Should the FA decide that the English candidates, Alan Pardew, Garry Monk, Gary Neville and Gareth Southgate, are not ready for the top job, then they will look at managers who have proved they know English football and English players.

The FA also hope that Hodgson’s successor will agree to work out of St George’s Park, rather than living in London and spending most of their time in the Wembley office.

Rodgers fits most of the criteria the FA look for in an England manager, having cut his coaching teeth at Chelsea and gained experience with Watford, Reading and Swansea City before landing the Liverpool job.

Harry Redknapp claims he wanted to take Rodgers as his assistant if he had been given the England job ahead of Hodgson.

At Anfield, Rodgers managed in Europe, almost won the Premier League title and got the best out of established stars such as Luis Suarez and Steven Gerrard, while restoring the confidence of Jordan Henderson and bringing through youngsters such as Raheem Sterling and Jordan Ibe.

Much will depend on whether or not Rodgers quickly returns to club management after being sacked last weekend, but if he is still free in the summer then Dyke will be attracted to the fact he claimed his “life’s work” is “trying to show that British players can play.”

Martinez also represents an intriguing option, as, despite the fact he was born in Balaguer in Spain, he spent the majority of his playing career in England and has only managed at Swansea, Wigan and Everton. The 42-year-old’s wife Beth is Scottish.

Watford’s Spanish manager Quique Flores this season said: “In Spain we think of Roberto as more like an English manager because he has had his complete career here. He gets results, so he is a good reference from Spanish and English football.”

Like Rodgers did at Liverpool, Martinez has a strong contingent of English players in his Everton squad with John Stones, Ross Barkley, Phil Jagielka, Leighton Baines, Tom Cleverley and 19-year-old Brendan Galloway all at Goodison Park.

Martinez has already claimed Stones is a future England captain and, while he has not managed in the Champions League, he has considerable Europa League experience.

As a Scot, Moyes may well be a tougher sell to the English public than Rodgers or Martinez, but that would not necessarily put off Dyke, who said: “You wouldn't rule out someone who is a Scot or Welsh or French, but they would have to understand English football.”

Moyes ticks as many boxes as Rodgers and Martinez, despite his unhappy spell at Manchester United.

It was Moyes who gave England captain Wayne Rooney his professional debut as a 16-year-old at Everton and the 52-year-old had a good record of giving English players, such as Jagielka and Baines, a chance at Goodison.

Although he was sacked after just 10 months in charge at Old Trafford, his time at United gives Moyes experience of managing one of the biggest clubs in the world.

The fact he has been prepared to try to resurrect his management career in Spain, with Real Sociedad, will also play well with Dyke and Glenn.

One man who would also count as ‘home-grown’ who is highly unlikely to be considered by the FA is Chelsea manager Jose Mourinho.

Mourinho rejected an approach to succeed Steve McClaren as England manager in 2007, but, aside from the fact he would be hugely expensive, the Portuguese has had so many run-ins with the FA that he is no longer thought to be a good fit.